3.153 \(\int \frac {(a+b \tan ^{-1}(\frac {c}{x}))^3}{x^3} \, dx\)

Optimal. Leaf size=147 \[ \frac {3 b^2 \log \left (\frac {2}{1+\frac {i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )}{c^2}+\frac {3 i b \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 c^2}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 x^2}+\frac {3 b \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c x}+\frac {3 i b^3 \text {Li}_2\left (1-\frac {2}{\frac {i c}{x}+1}\right )}{2 c^2} \]

[Out]

3/2*I*b*(a+b*arccot(x/c))^2/c^2+3/2*b*(a+b*arccot(x/c))^2/c/x-1/2*(a+b*arccot(x/c))^3/c^2-1/2*(a+b*arccot(x/c)
)^3/x^2+3*b^2*(a+b*arccot(x/c))*ln(2/(1+I*c/x))/c^2+3/2*I*b^3*polylog(2,1-2/(1+I*c/x))/c^2

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Rubi [F]  time = 2.25, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTan[c/x])^3/x^3,x]

[Out]

(((3*I)/64)*b^3*(1 - (I*c)/x)^2)/c^2 - (3*a*b^2*(1 + (I*c)/x)^2)/(16*c^2) - (((3*I)/64)*b^3*(1 + (I*c)/x)^2)/c
^2 - (((3*I)/8)*a^2*b)/x^2 - (3*a*b^2)/(8*x^2) + (3*a^2*b)/(4*c*x) - (3*b^3)/(2*c*x) + (((3*I)/4)*a^2*b*Log[I
- c/x])/c^2 + (3*a*b^2*Log[I - c/x])/(8*c^2) - (3*a*b^2*(1 - (I*c)/x)*Log[1 - (I*c)/x])/(4*c^2) + (((3*I)/4)*b
^3*(1 - (I*c)/x)*Log[1 - (I*c)/x])/c^2 + (3*a*b^2*Log[1 - (I*c)/x])/(8*x^2) - (3*b^2*(1 - (I*c)/x)^2*(2*a + I*
b*Log[1 - (I*c)/x]))/(32*c^2) + (((3*I)/8)*b*(1 - (I*c)/x)*(2*a + I*b*Log[1 - (I*c)/x])^2)/c^2 - (((3*I)/32)*b
*(1 - (I*c)/x)^2*(2*a + I*b*Log[1 - (I*c)/x])^2)/c^2 - ((1 - (I*c)/x)*(2*a + I*b*Log[1 - (I*c)/x])^3)/(8*c^2)
+ ((1 - (I*c)/x)^2*(2*a + I*b*Log[1 - (I*c)/x])^3)/(16*c^2) - (9*a*b^2*(1 + (I*c)/x)*Log[1 + (I*c)/x])/(4*c^2)
- (((3*I)/4)*b^3*(1 + (I*c)/x)*Log[1 + (I*c)/x])/c^2 + (3*a*b^2*(1 + (I*c)/x)^2*Log[1 + (I*c)/x])/(8*c^2) + (
((3*I)/32)*b^3*(1 + (I*c)/x)^2*Log[1 + (I*c)/x])/c^2 + (((3*I)/4)*a^2*b*Log[1 + (I*c)/x])/x^2 + (3*a*b^2*Log[1
+ (I*c)/x])/(8*x^2) - (3*a*b^2*Log[1 - (I*c)/x]*Log[1 + (I*c)/x])/(4*x^2) + (3*a*b^2*(1 + (I*c)/x)*Log[1 + (I
*c)/x]^2)/(4*c^2) + (((3*I)/8)*b^3*(1 + (I*c)/x)*Log[1 + (I*c)/x]^2)/c^2 - (3*a*b^2*(1 + (I*c)/x)^2*Log[1 + (I
*c)/x]^2)/(8*c^2) - (((3*I)/32)*b^3*(1 + (I*c)/x)^2*Log[1 + (I*c)/x]^2)/c^2 - ((I/8)*b^3*(1 + (I*c)/x)*Log[1 +
(I*c)/x]^3)/c^2 + ((I/16)*b^3*(1 + (I*c)/x)^2*Log[1 + (I*c)/x]^3)/c^2 + (3*a*b^2*Log[I + c/x])/(8*c^2) - (3*a
*b^2*Log[1 - (I*c)/x]*Log[c - I*x])/(4*c^2) - (3*a*b^2*Log[1 + (I*c)/x]*Log[c + I*x])/(4*c^2) + (3*a*b^2*Log[(
c - I*x)/(2*c)]*Log[c + I*x])/(4*c^2) + (3*a*b^2*Log[c - I*x]*Log[(c + I*x)/(2*c)])/(4*c^2) - (3*a*b^2*Log[c +
I*x]*Log[((-I)*x)/c])/(4*c^2) - (3*a*b^2*Log[c - I*x]*Log[(I*x)/c])/(4*c^2) + (3*a*b^2*PolyLog[2, (c - I*x)/(
2*c)])/(4*c^2) + (3*a*b^2*PolyLog[2, (c + I*x)/(2*c)])/(4*c^2) + (3*a*b^2*PolyLog[2, ((-I)*c)/x])/(4*c^2) + (3
*a*b^2*PolyLog[2, (I*c)/x])/(4*c^2) - (3*a*b^2*PolyLog[2, 1 - (I*x)/c])/(4*c^2) - (3*a*b^2*PolyLog[2, 1 + (I*x
)/c])/(4*c^2) + ((3*I)/8)*b^3*Defer[Int][(Log[1 - (I*c)/x]^2*Log[1 + (I*c)/x])/x^3, x] - ((3*I)/8)*b^3*Defer[I
nt][(Log[1 - (I*c)/x]*Log[1 + (I*c)/x]^2)/x^3, x]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 x^3}+\frac {3 i b \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 x^3}-\frac {3 i b^2 \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 x^3}+\frac {i b^3 \log ^3\left (1+\frac {i c}{x}\right )}{8 x^3}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{x^3} \, dx+\frac {1}{8} (3 i b) \int \frac {\left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right )^2 \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^2\right ) \int \frac {\left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {1}{8} \left (i b^3\right ) \int \frac {\log ^3\left (1+\frac {i c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac {1}{8} \operatorname {Subst}\left (\int x (2 a+i b \log (1-i c x))^3 \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{8} (3 i b) \int \left (-\frac {4 a^2 \log \left (1+\frac {i c}{x}\right )}{x^3}-\frac {4 i a b \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3}+\frac {b^2 \log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3}\right ) \, dx-\frac {1}{8} \left (3 i b^2\right ) \int \left (-\frac {2 i a \log ^2\left (1+\frac {i c}{x}\right )}{x^3}+\frac {b \log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3}\right ) \, dx-\frac {1}{8} \left (i b^3\right ) \operatorname {Subst}\left (\int x \log ^3(1+i c x) \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{8} \operatorname {Subst}\left (\int \left (-\frac {i (2 a+i b \log (1-i c x))^3}{c}+\frac {i (1-i c x) (2 a+i b \log (1-i c x))^3}{c}\right ) \, dx,x,\frac {1}{x}\right )\right )-\frac {1}{2} \left (3 i a^2 b\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{4} \left (3 a b^2\right ) \int \frac {\log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (i b^3\right ) \operatorname {Subst}\left (\int \left (\frac {i \log ^3(1+i c x)}{c}-\frac {i (1+i c x) \log ^3(1+i c x)}{c}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx\\ &=-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {1}{2} \left (3 i a^2 b\right ) \operatorname {Subst}\left (\int x \log (1+i c x) \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 a b^2\right ) \operatorname {Subst}\left (\int x \log ^2(1+i c x) \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1-\frac {i c}{x}\right )}{2 (c-i x) x^3} \, dx-\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1+\frac {i c}{x}\right )}{2 (c+i x) x^3} \, dx+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {i \operatorname {Subst}\left (\int (2 a+i b \log (1-i c x))^3 \, dx,x,\frac {1}{x}\right )}{8 c}-\frac {i \operatorname {Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^3 \, dx,x,\frac {1}{x}\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \log ^3(1+i c x) \, dx,x,\frac {1}{x}\right )}{8 c}-\frac {b^3 \operatorname {Subst}\left (\int (1+i c x) \log ^3(1+i c x) \, dx,x,\frac {1}{x}\right )}{8 c}\\ &=\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {1}{4} \left (3 a b^2\right ) \operatorname {Subst}\left (\int \left (\frac {i \log ^2(1+i c x)}{c}-\frac {i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {\operatorname {Subst}\left (\int (2 a+i b \log (x))^3 \, dx,x,1-\frac {i c}{x}\right )}{8 c^2}+\frac {\operatorname {Subst}\left (\int x (2 a+i b \log (x))^3 \, dx,x,1-\frac {i c}{x}\right )}{8 c^2}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \log ^3(x) \, dx,x,1+\frac {i c}{x}\right )}{8 c^2}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int x \log ^3(x) \, dx,x,1+\frac {i c}{x}\right )}{8 c^2}+\frac {1}{4} \left (3 a^2 b c\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+i c x} \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (3 a b^2 c\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{(c-i x) x^3} \, dx-\frac {1}{4} \left (3 a b^2 c\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{(c+i x) x^3} \, dx\\ &=-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {(3 i b) \operatorname {Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-\frac {i c}{x}\right )}{16 c^2}+\frac {(3 i b) \operatorname {Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-\frac {i c}{x}\right )}{8 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac {i c}{x}\right )}{16 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+\frac {i c}{x}\right )}{8 c^2}+\frac {\left (3 i a b^2\right ) \operatorname {Subst}\left (\int \log ^2(1+i c x) \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {\left (3 i a b^2\right ) \operatorname {Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,\frac {1}{x}\right )}{4 c}+\frac {1}{4} \left (3 a^2 b c\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {i x}{c}+\frac {i}{c^2 (-i+c x)}\right ) \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (3 a b^2 c\right ) \int \left (-\frac {i \log \left (1-\frac {i c}{x}\right )}{c^3 (c-i x)}+\frac {\log \left (1-\frac {i c}{x}\right )}{c x^3}+\frac {i \log \left (1-\frac {i c}{x}\right )}{c^2 x^2}-\frac {\log \left (1-\frac {i c}{x}\right )}{c^3 x}\right ) \, dx-\frac {1}{4} \left (3 a b^2 c\right ) \int \left (\frac {i \log \left (1+\frac {i c}{x}\right )}{c^3 (c+i x)}+\frac {\log \left (1+\frac {i c}{x}\right )}{c x^3}-\frac {i \log \left (1+\frac {i c}{x}\right )}{c^2 x^2}-\frac {\log \left (1+\frac {i c}{x}\right )}{c^3 x}\right ) \, dx\\ &=-\frac {3 i a^2 b}{8 x^2}+\frac {3 a^2 b}{4 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}-\frac {1}{4} \left (3 a b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{4} \left (3 a b^2\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-\frac {i c}{x}\right )}{16 c^2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}+\frac {\left (3 i a b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{c-i x} \, dx}{4 c^2}-\frac {\left (3 i a b^2\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{c+i x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+\frac {i c}{x}\right )}{16 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}-\frac {\left (3 i a b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{x^2} \, dx}{4 c}+\frac {\left (3 i a b^2\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x^2} \, dx}{4 c}\\ &=\frac {3 i b^3 \left (1-\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i a^2 b}{8 x^2}+\frac {3 a^2 b}{4 c x}-\frac {3 i a b^2}{2 c x}-\frac {3 b^3}{4 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{32 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{32 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}+\frac {1}{4} \left (3 a b^2\right ) \operatorname {Subst}\left (\int x \log (1-i c x) \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 a b^2\right ) \operatorname {Subst}\left (\int x \log (1+i c x) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+\frac {i c}{x}\right )}{2 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}+\frac {\left (3 i a b^2\right ) \int \frac {\log (c-i x)}{\left (1-\frac {i c}{x}\right ) x^2} \, dx}{4 c}-\frac {\left (3 i a b^2\right ) \int \frac {\log (c+i x)}{\left (1+\frac {i c}{x}\right ) x^2} \, dx}{4 c}+\frac {\left (3 i a b^2\right ) \operatorname {Subst}\left (\int \log (1-i c x) \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {\left (3 i a b^2\right ) \operatorname {Subst}\left (\int \log (1+i c x) \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=\frac {3 i b^3 \left (1-\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i a^2 b}{8 x^2}+\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{32 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{2 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{32 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}+\frac {\left (3 i a b^2\right ) \int \left (\frac {\log (c-i x)}{c (c+i x)}+\frac {i \log (c-i x)}{c x}\right ) \, dx}{4 c}-\frac {\left (3 i a b^2\right ) \int \left (\frac {\log (c+i x)}{c (c-i x)}-\frac {i \log (c+i x)}{c x}\right ) \, dx}{4 c}+\frac {1}{8} \left (3 i a b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-i c x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 i a b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+i c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3 i b^3 \left (1-\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i a^2 b}{8 x^2}+\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{32 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}-\frac {9 a b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{32 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {\left (3 i a b^2\right ) \int \frac {\log (c-i x)}{c+i x} \, dx}{4 c^2}-\frac {\left (3 i a b^2\right ) \int \frac {\log (c+i x)}{c-i x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log (c-i x)}{x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log (c+i x)}{x} \, dx}{4 c^2}-\frac {1}{8} \left (3 i a b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {i x}{c}+\frac {i}{c^2 (-i+c x)}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 i a b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}+\frac {i x}{c}-\frac {i}{c^2 (i+c x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {3 i b^3 \left (1-\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i a^2 b}{8 x^2}-\frac {3 a b^2}{8 x^2}+\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (i-\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{32 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}-\frac {9 a b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{32 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}+\frac {3 a b^2 \log \left (i+\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )}{4 c^2}-\frac {3 a b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right )}{4 c^2}-\frac {3 a b^2 \log (c-i x) \log \left (\frac {i x}{c}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {\left (3 i a b^2\right ) \int \frac {\log \left (\frac {c-i x}{2 c}\right )}{c+i x} \, dx}{4 c^2}+\frac {\left (3 i a b^2\right ) \int \frac {\log \left (\frac {c+i x}{2 c}\right )}{c-i x} \, dx}{4 c^2}+\frac {\left (3 i a b^2\right ) \int \frac {\log \left (-\frac {i x}{c}\right )}{c+i x} \, dx}{4 c^2}-\frac {\left (3 i a b^2\right ) \int \frac {\log \left (\frac {i x}{c}\right )}{c-i x} \, dx}{4 c^2}\\ &=\frac {3 i b^3 \left (1-\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i a^2 b}{8 x^2}-\frac {3 a b^2}{8 x^2}+\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (i-\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{32 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}-\frac {9 a b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{32 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}+\frac {3 a b^2 \log \left (i+\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )}{4 c^2}-\frac {3 a b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right )}{4 c^2}-\frac {3 a b^2 \log (c-i x) \log \left (\frac {i x}{c}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1-\frac {i x}{c}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1+\frac {i x}{c}\right )}{4 c^2}+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-i x\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+i x\right )}{4 c^2}\\ &=\frac {3 i b^3 \left (1-\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2}{64 c^2}-\frac {3 i a^2 b}{8 x^2}-\frac {3 a b^2}{8 x^2}+\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 i a^2 b \log \left (i-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (i-\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{32 c^2}+\frac {3 i b \left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{32 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{8 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^3}{16 c^2}-\frac {9 a b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{32 c^2}+\frac {3 i a^2 b \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {3 i b^3 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {3 i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{32 c^2}-\frac {i b^3 \left (1+\frac {i c}{x}\right ) \log ^3\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i b^3 \left (1+\frac {i c}{x}\right )^2 \log ^3\left (1+\frac {i c}{x}\right )}{16 c^2}+\frac {3 a b^2 \log \left (i+\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {3 a b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac {3 a b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )}{4 c^2}-\frac {3 a b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right )}{4 c^2}-\frac {3 a b^2 \log (c-i x) \log \left (\frac {i x}{c}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c-i x}{2 c}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c+i x}{2 c}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1-\frac {i x}{c}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1+\frac {i x}{c}\right )}{4 c^2}+\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log ^2\left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 i b^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 178, normalized size = 1.21 \[ \frac {a \left (a c (3 b x-a c)+6 b^2 x^2 \log \left (\frac {1}{\sqrt {\frac {c^2}{x^2}+1}}\right )\right )-3 b \tan ^{-1}\left (\frac {c}{x}\right ) \left (a \left (a \left (c^2+x^2\right )-2 b c x\right )-2 b^2 x^2 \log \left (1+e^{2 i \tan ^{-1}\left (\frac {c}{x}\right )}\right )\right )+3 b^2 (c-i x) \tan ^{-1}\left (\frac {c}{x}\right )^2 (b x-a (c+i x))-\left (b^3 \left (c^2+x^2\right ) \tan ^{-1}\left (\frac {c}{x}\right )^3\right )-3 i b^3 x^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}\left (\frac {c}{x}\right )}\right )}{2 c^2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c/x])^3/x^3,x]

[Out]

(3*b^2*(c - I*x)*(-(a*(c + I*x)) + b*x)*ArcTan[c/x]^2 - b^3*(c^2 + x^2)*ArcTan[c/x]^3 - 3*b*ArcTan[c/x]*(a*(-2
*b*c*x + a*(c^2 + x^2)) - 2*b^2*x^2*Log[1 + E^((2*I)*ArcTan[c/x])]) + a*(a*c*(-(a*c) + 3*b*x) + 6*b^2*x^2*Log[
1/Sqrt[1 + c^2/x^2]]) - (3*I)*b^3*x^2*PolyLog[2, -E^((2*I)*ArcTan[c/x])])/(2*c^2*x^2)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \arctan \left (\frac {c}{x}\right )^{3} + 3 \, a b^{2} \arctan \left (\frac {c}{x}\right )^{2} + 3 \, a^{2} b \arctan \left (\frac {c}{x}\right ) + a^{3}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c/x))^3/x^3,x, algorithm="fricas")

[Out]

integral((b^3*arctan(c/x)^3 + 3*a*b^2*arctan(c/x)^2 + 3*a^2*b*arctan(c/x) + a^3)/x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c/x))^3/x^3,x, algorithm="giac")

[Out]

integrate((b*arctan(c/x) + a)^3/x^3, x)

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maple [B]  time = 0.10, size = 396, normalized size = 2.69 \[ -\frac {a^{3}}{2 x^{2}}-\frac {b^{3} \arctan \left (\frac {c}{x}\right )^{3}}{2 x^{2}}-\frac {b^{3} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}+\frac {3 b^{3} \arctan \left (\frac {c}{x}\right )^{2}}{2 c x}-\frac {3 b^{3} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2 c^{2}}-\frac {3 i b^{3} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{4 c^{2}}-\frac {3 i b^{3} \ln \left (i+\frac {c}{x}\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{4 c^{2}}+\frac {3 i b^{3} \dilog \left (-\frac {i \left (i+\frac {c}{x}\right )}{2}\right )}{4 c^{2}}-\frac {3 i b^{3} \ln \left (i+\frac {c}{x}\right )^{2}}{8 c^{2}}-\frac {3 i b^{3} \dilog \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{4 c^{2}}+\frac {3 i b^{3} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (i+\frac {c}{x}\right )}{2}\right )}{4 c^{2}}+\frac {3 i b^{3} \ln \left (i+\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{4 c^{2}}+\frac {3 i b^{3} \ln \left (\frac {c}{x}-i\right )^{2}}{8 c^{2}}-\frac {3 a \,b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 x^{2}}-\frac {3 a \,b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}+\frac {3 a \,b^{2} \arctan \left (\frac {c}{x}\right )}{c x}-\frac {3 a \,b^{2} \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2 c^{2}}-\frac {3 b \,a^{2} \arctan \left (\frac {c}{x}\right )}{2 x^{2}}+\frac {3 a^{2} b}{2 c x}+\frac {3 a^{2} b \arctan \left (\frac {x}{c}\right )}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c/x))^3/x^3,x)

[Out]

-1/2*a^3/x^2-1/2/x^2*b^3*arctan(c/x)^3-1/2/c^2*b^3*arctan(c/x)^3+3/2/c*b^3*arctan(c/x)^2/x-3/2/c^2*b^3*arctan(
c/x)*ln(1+c^2/x^2)-3/4*I/c^2*b^3*ln(c/x-I)*ln(1+c^2/x^2)-3/4*I/c^2*b^3*ln(I+c/x)*ln(1/2*I*(c/x-I))-3/4*I/c^2*b
^3*dilog(1/2*I*(c/x-I))-3/8*I/c^2*b^3*ln(I+c/x)^2+3/4*I/c^2*b^3*ln(c/x-I)*ln(-1/2*I*(I+c/x))+3/4*I/c^2*b^3*ln(
I+c/x)*ln(1+c^2/x^2)+3/8*I/c^2*b^3*ln(c/x-I)^2+3/4*I/c^2*b^3*dilog(-1/2*I*(I+c/x))-3/2/x^2*a*b^2*arctan(c/x)^2
-3/2/c^2*a*b^2*arctan(c/x)^2+3/c*a*b^2/x*arctan(c/x)-3/2/c^2*a*b^2*ln(1+c^2/x^2)-3/2*b/x^2*a^2*arctan(c/x)+3/2
*a^2*b/c/x+3/2/c^2*a^2*b*arctan(x/c)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c/x))^3/x^3,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c/x))^3/x^3,x)

[Out]

int((a + b*atan(c/x))^3/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c/x))**3/x**3,x)

[Out]

Integral((a + b*atan(c/x))**3/x**3, x)

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